Why can't I build a non-reversible optic setup which focuses the sun to higher temperatures? Following this question I would like to challenge one of the assumptions.
The standard answer is that thermodynamics prohibits focusing the sun to a spot such that the spot reaches a higher temperature than the sun itself, because lenses and mirrors are reversible machines and the argument goes from there.
I don't see why the reversibility should hold for composite machines. As in, if I have one lens/mirror apparatus here which focuses the sun to, say, 5,500K, and a duplicate apparatus there which creates another such spot...
...and I tilt them so that the spots overlap...
...then intuitively the overlapping spot should have a temperature significantly higher than 5,500K...
...and the machine isn't reversible because a photon striking any given point could have taken either path, so the thermodynamic argument against the above result doesn't apply.
What's wrong with this reasoning?
 A: The ideal 5500K apparatus must produce perfect optical coupling between the spot and the sun. This will mean that all light emitted by the spot must be channeled back in the direction of the sun. No light emitted by the spot is allowed to be absorbed or escape into other directions in space.
With this in mind, the apparatus must surround the spot from all sides, so you cannot simply bring two such machines together.
A: I would explain this as follows:
The (hypothetical) apparatus that focuses sunlight to "5500K" will illuminate the target from the full half-space. So it cannot be combined with itself to generate higher photon flux.
Another way to put it is that you need many focusing apparatuses that illuminate the target from all sides to reach 5500 K in the first place.
I would relate this to the fact that the smallest spot size can only be achieved if the illumination is from the full half-space, or the optics has a maximum numerical aperture.
Does this make sense to you?
A: If one (let's call it a lens) optic focuses an image of the sun onto a target, then it does so, not from all directions (4pi steradians), but only from the
circle of the lens.   It is then possible to position another lens with
a clear line-of-sight to the object, and focus more sunlight from the second lens.
The limit is, after all 4pi steradians of the surroundings of the target
are filled with lenses focusing sunlight, you have finally coupled your target
with ONLY ONE object, the sun.  Until you completely surround the target,
the blank directions are presumably colder than the target and it is going
to radiate away in those directions, and not achieve solar equality. 
After you surround the target, you can no longer add one more lens.
One way a composite machine can exceed this limit, is to run a solar
steam engine, which drives a generator, which fires a spark plug.
The spark plug can achieve a higher temperature than the sun.
A: Don't forget the Lagrange Invariant. That's what it's all about. The sun has a finite size in object space and therefore has a field. Faster optical systems will provide smaller spot sizes of the sun but the angles will be larger. Longer focal length systems will provide larger images but the angles are smaller. As you squeeze the size down the angles go up, just like in diffraction. alpha = 2.44 lambda/diameter. Like in a projector, if you have a dim image you can't just put in a higher Wattage lamp because to get more Wattage, you must increase the size of the filament and a larger filament means a larger field of view for the optical system and won't get through the field stop.
